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NANO EXPRESS
Magnetic Anisotropic Energy Gap and Strain Effectin Au Nanoparticles
Po-Hsun Shih Æ Sheng Yun Wu
Received: 1 April 2009 / Accepted: 9 September 2009 / Published online: 22 September 2009
� to the authors 2009
Abstract We report on the observation of the size effect
of thermal magnetization in Au nanoparticles. The thermal
deviation of the saturation magnetization departs substan-
tially from that predicted by the Bloch T3/2-law, indicating
the existence of magnetic anisotropic energy. The results
may be understood using the uniaxial anisotropy Heisen-
berg model, in which the surface atoms give rise to
polarized moments while the magnetic anisotropic energy
decreases as the size of the Au nanoparticles is reduced.
There is a significant maximum magnetic anisotropic
energy found for the 6 nm Au nanoparticles, which is
associated with the deviation of the lattice constant due to
magnetocrystalline anisotropy.
Keywords Nanoparticles � Magnetic anisotropy �Magnetic properties � Spin waves
Introduction
Metal nanoparticles of Pd, Au, and Cu have been exten-
sively studied, because, due to a reduction in dimension-
ality, their ferromagnetic polarizations are quite different
from those observed in transition metals [1–6]. The most
frequent effects of the small size are lattice rearrangement,
crystalline imperfections, a higher degree of localization,
and narrowed valence band width. It has been reported in
previous studies [2, 4] that individual Pd and Au nano-
particles may reach their ferromagnetic moment at low
temperatures, and that, theoretically, there may be a slight
enhancement of the 4d localization, although Pd and Au are
both characterized by diamagnetism in the bulk state. Bulk
Au metal also demonstrates a typical diamagnetic response
of -1.42 9 10-6 emu/g [7], when the [Xe]4f145d106 s1 Au
configuration has a closed d shell and a single s electron.
Finite-size effects play a dominant role in determining the
magnetic properties. A decrease in size can lead to unusual
ferromagnetic and diamagnetic properties. The origin of
the ferromagnetism observed in filled 4d or 5d electron
nanoparticle systems can be explained as due to giant
magnetic anisotropy [8] and Fermi-hole effects [9] that
influence the evolution from the surface polarization spins
to the diamagnetic bulk state. In this letter, we discuss the
effects of surface polarization and weak magnetic anisot-
ropy in Au nanoparticles, which indicate the appearance of
ferromagnetic spin polarization and magnetic anisotropic
energy at low temperatures. Moreover, the strain induced
by the lattice can be used to tune the magnetic aniso-
tropic energy, which is obtained from the quantum spin
wave theory and the anisotropic Heisenberg ferromagnetic
model.
Experimental Details
The Au nanoparticles used in the present study were fab-
ricated by the thermal evaporation method. High-purity
gold ingots (99.999%) were evaporated in the range of 0.1–
2 T. The Ar gas was fed at a rate of *0.1 A/s. To avoid
contamination by magnetic impurities originating from the
stainless steel plate the samples were collected by a
rotating silicon substrate maintained at the temperature of
liquid nitrogen. The resultant samples consisted of collec-
tions of individual Au nanoparticles in the form of dried
powder. The morphology and structures of the prepared
P.-H. Shih � S. Y. Wu (&)
Department of Physics, National Dong Hwa University,
Hualien 97401, Taiwan
e-mail: [email protected]
123
Nanoscale Res Lett (2010) 5:25–30
DOI 10.1007/s11671-009-9438-z
nanoparticles were then characterized using transmission
electron microscopy (TEM, JEM-1400 JEOL).
Results and Discussion
Structural Analysis
It is clearly evident in the portion of the TEM images
shown in Fig. 1a that the nanoparticles are spherical and
well separated. The interconnecting nanoparticles are stuck
together in clusters due to electrostatic effects as well as an
artifact of the drying of aqueous suspensions. The size and
distribution of the nanoparticles can be calculated. An
examination of the portion of the TEM image shown in the
Fig. 1b clearly shows that size and distribution are quite
asymmetric and can be described using a log-normal dis-
tribution function. The log-normal distribution is defined as
follows: f ðdÞ ¼ 1ffiffiffiffi
2pp
drexp � ðln d�ln \d [ Þ2
2r2
� �
; where \d[ is
the mean value and r is the standard deviation of the
function. The mean diameter and standard deviation
obtained from the fits are \d[ = 3.7(9) nm and
r = 0.251, respectively. The small standard deviation
(r\ 0.5) of the function indicates that the distribution is
confined to a limited range. The broadening of the width of
the distribution profile is due to crystalline and nanoparticle
aggregation effects. The electron diffraction pattern cor-
responding to a selected area in the 3.7(9) nm Au nano-
particles is shown in Fig. 1c and clearly reveals the
crystalline nature of the sample. The pattern of the main
spots can easily be indexed as basically cubic in structure
with a space group of Fm-3m and a lattice parameter of
a = 4.07(4) A. This is consistent with earlier data for bulk
Au [7]. The diameters of the nanocrystals as determined
from TEM images of the samples used in this study were
approximately 3.7(9), 4.3(6), 5.6(4), 6.0(3), and 7.9(1) nm.
Magnetization
Magnetization measurements were performed using the
conventional superconducting quantum interference device
(Quantum Design, MPMS5) set up with magnetic fields
from -5 to 5 T, covering a temperature range from 5 to
300 K. The Au nanoparticle sample was mounted in a
sample holder capsule. Figure 2a shows the applied fields
and the resultant magnetization of Au nanoparticles (with a
mean diameter of 3.7(9) nm) obtained at the eight tem-
peratures. When a lower field is applied the magnetization
increases rapidly with the field; the increase follows a
curved path, revealing that the magnetization follows a
Langevin profile. At high temperatures, magnetization
saturation is reached at around Ha = 0.4 T which is fol-
lowed by a high-field linear decrease when the applied field
Ha reaches 1.2 T. There are no significant differences in
the magnetization measurements between the field-
increasing and the field-decreasing loops found above
25 K, which are consistent with the superparamagnetic
behaviors. Figure 2b shows the representative M(T) curves
taken for the 3.7(9) nm Au nanoparticle assemblies at the
selected applied magnetic field Ha = 0.5 T, revealing a
superparamagnetic behavior that can be described by
temperature dependence Langevin function. The resultant
Fig. 1 a TEM images of Au
nanoparticles; b size
distribution obtained from a
portion of a TEM image of Au
nanoparticles, which can be
described using a log-normal
distribution function, as
indicated by the solid curve;
c electron diffraction patterns of
a selected area 3.7(9) nm Au
nanoparticles, revealing the
cubic structure of the Au
nanoparticles
26 Nanoscale Res Lett (2010) 5:25–30
123
fitting parameters are shown in Fig. 2b. This suggests
the existence of two different magnetic components in
the sample—a superparamagnetic and a diamagnetic
component. A representative hysteresis loop taken at 5 K is
shown in Fig. 3. A distinguishable asymmetric coercivity
Hc = 175 Oe can be observed in the low Ha regime, which
signals the existence of ferromagnetic spin in the 3.7(9) nm
Au nanoparticles. The value obtained for coercivity is close
to other previously published values for similar Au nano-
particles [10]. The asymmetric characteristics are assumed
to originate from competition between the unidirectional
and uniaxial anisotropy [11, 12].
Consequently, we can describe the superparamagnetic
system using a Langevin function in combination with a
linear component associated with diamagnetism [13, 14].
The resultant total magnetization can be expressed as
MðH; TÞ ¼ MsLðxÞ þ vDH; x ¼lpH
kBT: ð1Þ
Here L(x) = coth(x)-1/x is the Langevin function, Ms is
the saturation magnetization, kB is the Boltzmann’s con-
stant, and vD is the diamagnetic susceptibility term. The
analysis relevant to Eq. 1 is based on a model which
ignores the inter-particle interactions and the contributions
of the distributions of the magnetic moment due to the log-
normal size distribution of the nanoparticle system [15]. It
can be seen that the fitted curves (solid line in Fig. 2a) are
quite consistent with the experimental data. The mecha-
nism often invoked to explain the occurrence of surface-
spin polarization effects in nonmagnetic particles [4] is that
the shell of the particle is ordered as a ferromagnetic shell,
while the core of each Au nanoparticle still behaves as a
diamagnetic single domain. Indeed, there is a discrepancy
between the data and the Langevin profile shown in the
M(H) curves taken at the low field regime. One possible
cause of this difference in the fit is the production of a
nonmagnetic surface layer by the chemical interaction
between the particle and the oxidation. In light of the
results obtained in various studies [16], we believe that this
difference has a different origin. An alternate explanation
has been made by Berkowitz [17, 18], who attributed the
reduction in the expected magnetization at low temperature
to difficulty in reaching saturation, because of a large
surface anisotropy.
Magnetic Anisotropic Energy Gap
The thermal deviation of the saturation magnetization can
be used to identify the anisotropic energy gap. Figure 4
shows the dependency of the thermal energy of the thermal
deviation DMs(T) = [Ms(5K)-Ms(T)]/Mth on the saturation
magnetization, as obtained from the fitting of Eq. 1, where
Mth is the saturated magnetization taken at room temper-
ature. The DMS(T) curve follows the Bloch T3/2-law
(dashed line) expected for ferromagnetic isotropic systems
below 10 K [14, 19] but departs from the curve in the high
Fig. 2 a Effects of the various temperatures plotted in relation to the
magnetization. The solid lines represent the fitted results. b Temper-
ature dependence of the M(T) curve taken for the 3.7(9) nm Au
nanoparticle assemblies at the selected applied magnetic field
Ha = 0.5 T
Fig. 3 Magnetization loops at 5 K for the 3.7(9) nm Au nanoparti-
cles revealing the appearance of magnetic hysteresis in the low field
regime
Nanoscale Res Lett (2010) 5:25–30 27
123
thermal energy regime, signaling the onset of magnetic
anisotropy [15, 20–23], presumably due to the high sur-
face-to-volume ratio of the nanoparticles. The discrepancy
of fit above 10 K (for low applied fields) may be associated
with the effects of uniaxial anisotropy [20–22] and with
inhomogeneities in the magnetic moments [23], which
have been ignored in the above analysis.
Here, we consider the surface and anisotropic effects in
a quantum spin wave model for the Heisenberg ferro-
magnetic model [24, 25]. We can incorporate the spin–spin
effects and anisotropy between coupling constants, that are
known to be important in a nano-size system, into the
anisotropic Hamiltonian, but do not include diamagnetic
effects [26–28]
H ¼ �X
i;j
½JzSizSjz þ JxyðSixSjx þ SiySjyÞ� � mBX
i
Siz;
ð2Þ
where the sum in the first term is the anisotropic ferromag-
netic Heisenberg exchange interaction (Jz and Jxy) between
nearest-neighbor spins on a nanoparticle; S denotes the spin
component of the electrons; and the last part corresponds to
the Zeeman energy (mS is the magnetic moment per atom).
The theory and the method of calculation have already
been described in detail elsewhere [26–28], therefore only
a few basic steps will be given here. We utilize an external
perturbation method and calculate the energy in the ground
state of the spin wave with wave vector q and dispersion
relation e0(q). We can now rewrite the equation as
e0ðqÞ ¼ SX
d
ðJz � Jxyeiq�dÞ; ð3Þ
where d is the nearest-neighbor vector. The lattice
constants for Au face-centered cubic nanoparticles with
Fm-3m symmetry are a = 4.07(4) A, with a nearest-
neighbor spacing of �a ¼ a=ffiffiffi
2p
: The dispersion relation can
be rewritten as
e0ðqÞ ¼ 12SðJz � JxyÞ þ 12SJxy
�
1 � 1
6
�
cos qx�a cos qy�a
þ cos qx�a cos qz�a þ cos qy�a cos qz�a�
�
: ð4Þ
This anisotropy in the coupling constants produces an
energy gap in the spin wave spectrum of D = 12S(Jz -
Jxy). The gap leads to an exponential dependence of the
order parameter on the thermal energy kBT:
DMSðTÞ ¼ MSð5KÞ � MSðTÞ½ �=Mth � e� D
kBT : ð5Þ
The solid lines indicate the results from the fit of Eq. 5; the
fitting parameters are listed in Table 1; the energy gap
obtained from the fit is plotted with the diameter in the
Fig. 5a (right panel). In the case of 3.7(9) nm, at higher
thermal energy kBT * 8 meV, the monotonic change of
DLs is closed to one and will be overcame by the thermal
energy. The direction of the magnetization of each Au
nanoparticle simply follows the direction of the applied
magnetic field. Consequently, the magnetization becomes
superparamagnetic and shows paramagnetic properties.
Fig. 4 Plot of the dependency of the thermal energy on the saturation
magnetization Ms(T) together with the thermal deviation DMs(T) due
to the saturation magnetization. The solid lines represent the fitted
results
Table 1 Summary of the size and fitting results for Au nanoparticles
\d[ (nm) Ms(5 K, emu/g) Lth(emu/g) D(meV)
3.7(9) 0.115 0.015 0.646
4.3(6) 0.058 0.004 0.709
5.6(4) 0.031 0.0085 2.391
6.0(3) 0.065 0.0076 6.527
7.9(1) 0.014 0.0032 1.412
Fig. 5 a Plots of the variation in D and strain e with mean diameter,
revealing the increase in magnetic anisotropic energy with increasing
particle size. b Schematic plots for negative strain (e\ 0), and cpositive strain (e[ 0)
28 Nanoscale Res Lett (2010) 5:25–30
123
The larger the size of the nanoparticle, the higher the
magnetic anisotropic energy, which therefore increases
with increasing particle size, until reaching the maximum
magnetic anisotropic energy: D = 6.527 meV in the
6.0(3) nm Au nanoparticles. The results are in good
agreement with the molecular field theories, which
predict linear or exponential variations for large and
small anisotropic energies, depending on whether a
classical or quantized system is used for the magnetic
moment [29]. In general, magnetic anisotropy means the
dependence of the internal energy of a system on the
direction of the spontaneous magnetization. Most kinds of
magnetic anisotropy are related to the deviations in the
lattice constant of the strain, known as magnetocrystalline
anisotropy [30]. Figure 5 shows the strain as a function of
mean diameter \d[. Shown in Fig. 5b (left panel), the
relative strain can be estimated from the change in the
a-axis lattice constant of Au nanoparticles
eð%Þ ¼ a � a0
a0
ð�100%Þ; ð6Þ
where a and a0 (4.076 A for bulk Au) indicate the lattice
constants of the strained and unstrained crystal, respec-
tively. In general, the spin–orbit interaction will induce a
small orbital momentum, which couples the magnetic
moment to the crystal axes. In a negative strained nano-
crystalline system, the wavefunctions between neighboring
atoms will overlap and reduce the magnetic anisotropy.
A reduction in the size of the nanoparticles (e\ 0) results
in unit cell contraction, which increases the stability of the
higher symmetry lattice and the coupling strength of
wavefunctions, shown in Fig. 5b. In a positive strained
e [ 0 nanocrystalline system, shown in Fig. 5c, the lattice
expands and decreases the coupling of wavefunctions. In
the case of our Au nanoparticles, the maximum positive
deviation in strain was observed when the mean size was
6.0(3) nm. The tendency of strain of size effects was
similar with the results of anisotropic energy. However, one
possible explanation for the higher strain state accompany
with higher magnetic anisotropy energy is an indicative of
lattice- and magnetic-anisotropy for Au nanoparticles.
Conclusions
An analysis of the results leads to an interesting conclu-
sion: that nanosized transition metal Au particles exhibit
both ferromagnetism and superparamagnetism, which are
in contrast to the metallic diamagnetism characteristic of
bulk Au. The superparamagnetic component of Au nano-
particles shows an anomalous temperature dependence that
can be well explained by the modified Langevin function
theory. Weak magnetic anisotropy was observed in the
mean deviation magnetization. The energy of the magnetic
anisotropic can be determined from the fitting of the
anisotropic Heisenberg model and related with the change
of strain. One possible explanation for the origin of the
observed superparamagnetic component of the magnetiza-
tion would be the existences of non-localized holes and
charge transfer which would signify that deviation from
stoichiometry would make only a small paramagnetic con-
tribution to the magnetization [31].
Acknowledgments We appreciate the financial support of this
research from the National Science Council of the Republic of China
under grant No. NSC-97-2112-M-259-004-MY3.
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